Regime-Switching Bivariate Dual Change Score Model

Abstract

Mixture structural equation model with regime switching (MSEM-RS) provides one possible way of representing over-time heterogeneities in dynamic processes by allowing a system to manifest qualitatively or quantitatively distinct change processes conditional on the latent “regime” the system is in at a particular time point. Unlike standard mixture structural equation models such as growth mixture models, MSEM-RS allows individuals to transition between latent classes over time. This class of models, often referred to as regime-switching models in the time series and econometric applications, can be specified as regime-switching mixture structural equation models when the number of repeated measures involved is not large. We illustrate the empirical utility of such models using one special case—a regime-switching bivariate dual change score model in which two growth processes are allowed to manifest regime-dependent coupling relations with one another. The proposed model is illustrated using a set of longitudinal reading and arithmetic performance data from the Early Childhood Longitudinal Study, Kindergarten Class of 1998–99 study (ECLS-K; U.S. Department of Education, National Center for Education Statistics, 2010).

Notes

¹For instance, with two classes at each time point and three time points, there is a total of 2³ = 8 possible ways in which individual's i class membership may vary over time. Thus, h indexes one specific pattern of class history out of eight possible patterns.

²Specifically, one plus the self-feedback parameter for a process results in the lag-1 autoregression coefficient of the process on itself. For details, see the formulation detailed in Chow, Ho, Hamaker, and Dolan (2010).

³We also considered a three-regime model, but the two-regime model with covariates already took approximately 1.46 hr to run on an Intel(R) Core(TM) 2.39 GHz 2-quad computer with 3.00 GB of RAM when specified with 10 sets of random starting values in the initial stage and 3 random sets of starting values in the final stage. The three-regime model also did not converge. Due to the lack of clear theoretical reason for positing a three-regime model over other alternative models with fewer regimes, we did not consider the three-regime model further.

^aGiven that we allowed the transition probability matrix at t = 2 to differ from subsequent transition probability matrix at later time points, the two regimes that existed at Time 1 were not necessarily the same as the two regimes (i.e., the Coupled and Decoupled regimes) extracted from other later time points.

⁴The logit intercept parameter, a ₁₂, was not significantly different from zero. Setting this parameter to zero in the best-fitting model led to an equal probability of transitioning from the low performance class into any of the two regimes at t = 2.

⁵For instance, with only two regimes and T = 9, there are 2⁹ = 512 ways in which the participants' regime history is manifested over time. The increase in computational time was substantial when the number of time points in our proposed regime-switching BDCS model was increased from five to nine. In the former, a single Monte Carlo replication with five sets of random starting values in the initial stage and two sets of random starting values in the second stage took approximately 1 min to run. When the number of time points was increased to nine (with missing values), a single replication with 10 random starting values in the initial stage and 3 sets of random starting values in the second stage took approximately 1.5 hr to run.